The n-Category Café

I just want to point out that “Jeremy Bentham’s Head” would make a great band name.

I did my postgrad study at UCL — his head is kept in a vacuum-sealed glass jar which, bizarrely enough, used to sit in-between his feet. It’s now kept locked away in the college vaults.

They had a picture of this old arrangement pinned up on the box a few years ago when Jeremy had been taken away to have his straw matting replaced (it had become infested). The collected ensemble looked slightly ghoulish to say the least!

Later I heard (and this may well be apocryphal) that the reason the head was taken off display in the case was that it had been kidnapped by a group of gadabouts from King’s College London as a merry rag-week stunt. One version of the story has them playing football with it (surely not true!)

Yargh! Found an image of JB’s (Jeremy Bentham I mean!) auto-icon complete with head!

JB assured me

Everyone else told me he was mummified.

Noooo! Not another thing I know that ain’t so! At this rate, my quest for knowledge will reach completion when I know nothing at all except that I know nothing. What a dismal prospect!

Tim wrote:

Noooo! Not another thing I know that ain’t so!

Wait a minute there — don’t jump off that bridge yet! You could be right — it’s possible the people I knew just took one look at Bentham’s corpse and thought “mummy”, without bothering to investigate precisely how it reached its current condition.

He’s listed in the Wikipedia article on self-mummification:

In the 1830s, Jeremy Bentham, the founder of utilitarianism, left instructions to be followed upon his death which led to the creation of a sort of modern-day mummy. He asked that his body be displayed to illustrate how the “horror at dissection originates in ignorance”; once so displayed and lectured about, he asked that his body parts be preserved, including his skeleton (minus his skull, for which he had other plans), which were to be dressed in the clothes he usually wore and “seated in a Chair usually occupied by me when living in the attitude in which I am sitting when engaged in thought.” His body, outfitted with a wax head created because of problems preparing it as Bentham requested, is on open display in the University College London.

But, does this really rule out the possibility that his innards were removed and his body stuffed? I don’t know.

The drawing at the lower right of the second page of the PDF is isomorphic to a pentatope . The comment on “not true in 4D” relates also to polyhedra able to be triangulated, but not always tetrahedralized.

n-category diagrams drawn on the shimmering iridescent surfaces of soap bubbles. I really want to that in full-color high-res video!

Didn’t you show us the soap bubble 120-cell?

They’re not quite the answer to your question, but I urge you to learn about ‘fake surfaces’.

It’s an answer to the anything else part of my question. So thanks.

Fake surfaces are just what you need for drawing the 2-morphisms in 2Vect that you can build from a ‘semisimple 2-algebra’.

Is there a more generic way of saying where they’re useful? E.g., just like string diagrams for adjunctions are not just relevant to adjunctions between categories, but also to adjunctions in any bicategory.

How could we know that there isn’t a further interesting class of surface, called perhaps ‘pseudo-fake surfaces’, corresponding to algebraically interesting things occurring?

Will novel things emerge in higher dimensions, or will it just be about coherence relations of constructions from lower dimensions?

John wrote:

Fake surfaces are just what you need for drawing the 2-morphisms in 2Vect that you can build from a ‘semisimple 2-algebra’.

David wrote:

Is there a more generic way of saying where they’re useful?

Good point — why limit ourselves to the world of linear algebra?

Probably fake surfaces are 2-morphisms in something like the ‘free symmetric monoidal 2-category on a symmetric Frobenius 2-algebra’.

A Frobenius algebra sounds like something in the world of linear algebra, but it’s actually something we can define in any monoidal category: we get an ordinary Frobenius algebra when we take that monoidal category to be Vect. You’ve seen what kind of pictures correspond to morphisms in the ‘free monoidal category on a Frobenius algebra’ — they’re in week224.

But this is just part of a big periodic table of Frobenius gadgets!

The best introduction so far may be this:

Aaron Lauda, Frobenius algebras and planar open string topological field theories, available as
math.QA/0508349.

After considering some baby cases, he draws the pictures for 2-morphisms in the ‘free monoidal 2-category on a Frobenius 2-algebra’. (He doesn’t use quite the same jargon, but never mind.)

Going symmetric would add extra wiggle room.

I’m not sure I’ve got the details exactly right, but fake surfaces will certainly be the 2-morphisms in a nice ‘free blah-di-blah on a blee-di-blee’, and I think I’m pretty close.

The main thing is that the tensor product in our 2-algebra gives us surfaces where 3 half-planes meet along a line, while the associator gives us surfaces where 4 half-planes meet at a vertex. The ‘Frobeniusness’ means we can freely switch inputs and outputs — it’s a way of getting our 2-algebra to be its own dual.

Thanks, John, your suggestion worked! Well, I also had to re-boot a few times when the Flash plug-in seemed to corrupt the OS at various points later on, but I managed to see the whole thing by skipping past the trouble spots each time I started again.

Anyway, it was definitely worth the effort!

I’m a bit surprised everyone doesn’t already know about Outside In. If you didn’t know about it, you might like the other of the Geometry Center’s videos, Not Knot (and part 2 of same).

John A. wrote:

I’m a bit surprised everyone doesn’t already know about Outside In.

I’m a bit surprised everyone doesn’t already know about $n$-categories! But as long as there are guys holed up near the Afghanistan–Pakistan border who don’t know and love everything about math and physics that I do, I’m gonna keep writing This Week’s Finds and dropping it in leaflet form out of helicopter windows worldwide. I’ll get you eventually, Osama!

Greg writes:

Thanks, John, your suggestion worked! Well, I also had to re-boot a few times when the Flash plug-in seemed to corrupt the OS at various points later on, but I managed to see the whole thing by skipping past the trouble spots each time I started again.

Anyway, it was definitely worth the effort!

Wow, that’s quite a compliment — it must have been good if it was worth that much effort. Corrupting the operating system?? What Flash plug-in are you talking about? I didn’t even know Flash was needed to watch this stuff. I’d like to make the technical prerequisites as low as possible.

Maybe I should learn to convert .mov files to some other formats…

Greg: do you think your problems could have been caused by bit errors in the downloaded file?

John said:

Anyway, thanks a million, and I hope you continue to correct me as I continue to blunder like a drunken bull into this beautiful china shop.

Well, I had to redeem myself after that embarrassment about the non-naturality of the map from direct sums to direct products.

This is the only point where I feel compelled to disagree. There may not be a pure-thought answer yet, but the integers are too important for us to rest content until we have a simple story about why their spectrum is 3-dimensional. As long as we say ‘oh, it’s just a complicated calculation’, we won’t get the gut-level intuitive understanding needed to make certain kinds of progress.

I agree completely. And it’s good to see you’re turning into a *real* number theorist who’s interested in $\mathbf{Z}$, and not one of those wimpy guys like Drinfeld and Lafforgue who study function fields. :)

Here is a reason on a very gut level. First consider a manifestly geometric example. A point has dimension 0, a punctured disk has dimension 1, and a Riemann surface has dimension two. Going up in dimension comes first from local winding and then globalization.

Similarly, a finite field has dimension one. A local field, such as $\mathbf{Q}_p$ or Laurent series $\mathbf{F}_q((x))$ has one higher dimension, the extra dimension coming from local winding, or ”ramification” as those in the business call it. A ring of integers in a global field, for example $\mathbf{Z}$ or $\mathbf{F}_q[x]$, has one more dimension coming from globalization.

So if you allow yourself to think of $\mathbf{Z}$ as being the ring of functions on something like a Riemann surface, then it’s all the usual stuff except that now your points are 1-dimensional, so everything else gets bumped up by one.

I don’t know if you like that better (or worse, for that matter).

Conc.: “absent a (non-formal) theory of the field with one element,”

Harans article looks very interesting.

My friend the number theorist Minhyong Kim has a new blog where he answers math questions, so I posted a question about why Spec($\mathbb{Z}$) is 3-dimensional, and he posted an interesting reply.

David Corfield (and others) might also like Minhyong’s essay on Mathematical Vistas.

From what I remember, this is not true, though I don’t have a ready counter-example. But it is true locally: the ring of integers in any finite extension of the field of $p$-adic numbers is a homomorphic image of $Z_p[x]$, where $Z_p$ denotes the ring of $p$-adic integers and where $p$ is a prime number.

Picturing the affine line over $Z$ (i.e. Spec $Z[x]$) from this topological point of view might be pushing the analogy beyond its breaking point. It’s tempting to view it as a family of affine lines parametrized by Spec $Z$—in other words, by our 3-manifold. But the topological affine line is just the usual space $C$ of complex numbers, so this fibration would have the same homotopy type as the 3-manifold. While I don’t know for a fact that $Z[x]$ and $Z$ don’t have the same etale homotopy type, it seems a bit unlikely to me. For instance for any finite field $F$, the rings $F[x]$ and $F$ don’t have the same etale homotopy type: because of wild ramification, the affine line isn’t simply connected in characteristic $p$!

On the other hand, I would bet that for the projective line $P^1$ over $Z$, the analogy is as good as the original one was. Then $P^1$ would be interpreted as a space fibered in 2-spheres (=topological projective lines) over the 3-manifold that is supposed to be Spec $Z$.

James wrote:

While I don’t know for a fact that $\mathbb{Z}[x]$ and $\mathbb{Z}$ don’t have the same etale homotopy type, it seems a bit unlikely to me.

I don’t know anything about this stuff, but could this issue be related to $\mathrm{A}^1$ homotopy theory, where people try to use the affine line as the parameter space for a new kind of ‘homotopy’, applicable to algebraic geometry?

A bit more precisely: could it be that $\mathbb{Z}$ and $\mathbb{Z}[x]$ fail to be homotopy equivalent in etale homotopy theory, but become so in $\mathrm{A}^1$ homotopy theory?